Related papers: Mixed multiplicities of arbitrary modules

Let $R$ be a commutative Noetherian ring and $M$ be a finitely generated $R$-module. Considering the new concept of linkage of ideals over a module, we study associated prime ideals, cofiniteness and Artinianness of local cohomology modules…

In this paper we study standard bases for submodules of a mixed power series and polynomial ring $R[[t_1,\ldots,t_m]][x_1,\ldots,x_n]^s$ respectively of their localization with respect to a $t$-local monomial ordering for a certain class of…

Using a recent Furstenberg structure theorem, we obtain a quantitative multiple recurrence theorem relative to any locally compact second countable Noetherian module over a syndetic ring.

We prove the Mixing Conjecture of Michel--Venkatesh for the class group action on Heegner points of large discriminant on compact arithmetic surfaces attached to maximal orders in rational quaternion algebras. The proof is conditional on…

Let $\mathfrak{a}$ be an ideal of a noetherian (not necessarily local) ring $R$ and $M$ an $R$-module with $\mathrm{Supp}_RM\subseteq\mathrm{V}(\mathfrak{a})$. We show that if $\mathrm{dim}_RM\leq2$, then $M$ is $\mathfrak{a}$-cofinite if…

The Buchsbaum-Eisenbud-Horrocks rank conjecture proposes lower bounds for the Betti numbers of a graded module M based on the codimension of M. We prove a special case of this conjecture via Boij-Soederberg theory. More specifically, we…

In this paper we show that the Rees algebra can be made into a functor on modules over a ring in a way that extends its classical definition for ideals. The Rees algebra of a module M may be computed in terms of a "maximal" map f from M to…

Let $R$ be a Noetherian ring. For a finitely generated $R$-module $M$, Northcott introduced the reducibility index of $M$, which is the number of submodules appearing in an irredundant irreducible decomposition of the submodule $0$ in $M$.…

We generalize the notion of and results on maximal proper quadratic modules from commutative unital rings to $\ast$-rings and discuss the relation of this generalization to recent developments in noncommutative real algebraic geometry. The…

We introduce a new invariant for subcategories X of finitely generated modules over a local ring R which we call the radius of X. We show that if R is a complete intersection and X is resolving, then finiteness of the radius forces X to…

Using invariants from commutative algebra to count geometric objects is a basic idea in singularities. For example, the multiplicity of an ideal is used to count points of intersection of two analytic sets at points of non-transverse…

We introduce the notion of a lim Cohen-Macaulay sequence of modules. We prove the existence of such sequences in positive characteristic, and show that their existence in mixed characteristic implies the long open conjecture about…

In this article, the authors introduce a class of mixed-norm Herz spaces, $\dot{E}^{\vec{\alpha},\vec{p}}_{\vec{q}}(\mathbb{R}^{n})$, which is a natural generalization of mixed Lebesgue spaces and some special cases of which naturally…

For a reduced Noetherian ring $R$ of characteristic $p > 0$, in this paper we discuss an extension of $R$ called its perfect closure $R^\infty$. This extension contains all $p^e$-th roots of elements of $R$, and is usually non-Noetherian.…

The acquisition of the defining equations of Rees algebras is a natural way to study these algebras and allows certain invariants and properties to be deduced. In this paper, we consider Rees algebras of codimension 2 perfect ideals of…

The purpose of this paper and its sequel, is to introduce a new class of modules over a commutative ring $R$, called $\mathbb{P}$-radical modules (modules $M$ satisfying the prime radical condition "$(\sqrt[p]{{\cal{P}}M}:M)={\cal{P}}$" for…

Recent results of Kahle and Miller give a method of constructing primary decompositions of binomial ideals by first constructing "mesoprimary decompositions" determined by their underlying monoid congruences. These mesoprimary…

Let $R$ be a commutative Noetherian local ring. We prove that the finiteness of the injective dimension of a finitely generated $R$-module $C$ is determined by the existence of a Cohen--Macaulay module $M$ that satisfies an inequality…

In this paper, we study the $F$-rationality of the Rees algebra and the extended Rees algebra of $\mathfrak{m}$-primary ideals in excellent local rings $(R, \mathfrak{m})$ of prime characteristic. We partially answer some conjectures and…

The Multiplicity conjecture of Herzog, Huneke, and Srinivasan states an upper bound for the multiplicity of any graded $k$-algebra as well as a lower bound for Cohen-Macaulay algebras. In this note we extend this conjecture in several…